scholarly journals Hilary Putnam on the philosophy of logic and mathematics

2018 ◽  
Vol 33 (2) ◽  
pp. 183
Author(s):  
José Miguel Sagüillo Fernández-Vega
Keyword(s):  
Final Section ◽  
Mathematical Practice ◽  
Logical Truth ◽  
Hilary Putnam ◽  
Philosophy Of Logic ◽  
Logical Validity ◽  
And Mathematics

I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.

Philosophy of Logic. Hilary Putnam

1973 ◽  
Vol 40 (1) ◽  
pp. 131-133
Author(s):  
John Corcoran
Keyword(s):  
Hilary Putnam ◽  
Philosophy Of Logic

scholarly journals O ENSINO E A PESQUISA SOBRE POLÍTICA EXTERNA NO CAMPO DAS RELAÇÕES INTERNACIONAIS DO BRASIL

2012 ◽  
Vol 1 (2) ◽  
pp. 95-128 ◽  
Author(s):  
Carlos Aurélio Pimenta de Faria
Keyword(s):  
Social Sciences ◽  
Foreign Policy ◽  
International Relations ◽  
Final Section ◽  
Graduate Programs ◽  
Fourth Section ◽  
The Third ◽  
The Status ◽  
Theses And Dissertations

The purpose of this article is to analyze teaching and research on foreign policy in Brazil in the last two decades. The first section discusses how the main narratives about the evolution of International Relations in Brazil, considered as an area of knowledge, depict the place that has been designed, in the same area, to the study of foreign policy. The second section is devoted to an assessment of the status of foreign policy in IR teaching in the country, both at undergraduate and scricto sensu graduate programs. There is also a mapping and characterization of theses and dissertations which had foreign policy as object. The third section assesses the space given to studies on foreign policy in three academic forums nationwide, namely: the meetings of ABRI (Brazilian Association of International Relations), the ABCP (Brazilian Association of Political Science) and ANPOCS (National Association of Graduate Programs and Research in Social Sciences). In the fourth section there is a mapping and characterization of the published articles on foreign policy between 1990 and 2010, in the following IR Brazilian journals: Cena Internacional, Contexto Internacional, Política Externa and Revista Brasileira de Política Internacional. At last, the fifth and final section seeks to assess briefly the importance that comparative studies have in the sub-area of foreign policy in the country. The final considerations make a general assessment of the empirical research presented in the previous sections.

Bradley, Francis Herbert (1846–1924)

2018 ◽  
Author(s):  
Stewart Candlish
Keyword(s):  
Philosophy Of History ◽  
Turn Of The Century ◽  
Bertrand Russell ◽  
General Interest ◽  
Philosophy Of Logic ◽  
Ultimate Truth ◽  
Absolute Idealism ◽  
Grammatical Subject ◽  
The World ◽  
And Mathematics

Bradley was the most famous and philosophically the most influential of the British Idealists, who had a marked impact on British philosophy in the later nineteenth and earlier twentieth centuries. They looked for inspiration less to their British predecessors than to Kant and Hegel, though Bradley owed as much to lesser German philosophers such as R.H. Lotze, J.F. Herbart and C. Sigwart. Bradley is most famous for his metaphysics. He argued that our ordinary conceptions of the world conceal contradictions. His radical alternative can be summarized as a combination of monism (that is, reality is one, there are no real separate things) and absolute idealism (that is, reality is idea, or consists of experience – but not the experience of any one individual, for this is forbidden by the monism). This metaphysics is said to have influenced the poetry of T.S. Eliot. But he also made notable contributions to philosophy of history, to ethics and to the philosophy of logic, especially of a critical kind. His critique of hedonism – the view that the goal of morality is the maximization of pleasure – is still one of the best available. Some of his views on logic, for instance, that the grammatical subject of a sentence may not be what the sentence is really about, became standard through their acceptance by Bertrand Russell, an acceptance which survived Russell’s repudiation of idealist logic and metaphysics around the turn of the century. Russell’s and G.E. Moore’s subsequent disparaging attacks on Bradley’s views signalled the return to dominance in England of pluralist (that is, non-monist) doctrines in the tradition of Hume and J.S. Mill, and, perhaps even more significantly, the replacement in philosophy of Bradley’s richly metaphorical literary style and of his confidence in the metaphysician’s right to adjudicate on the ultimate truth with something more like plain speaking and a renewed deference to science and mathematics. Bradley’s contemporary reputation was that of the greatest English philosopher of his generation. This status did not long survive his death, and the relative dearth of serious discussion of his work until more general interest revived in the 1970s has meant that the incidental textbook references to some of his most characteristic and significant views, for example, on relations and on truth, are often based on hostile and misleading caricatures.

Logical constants

2018 ◽  
Author(s):  
Timothy McCarthy
Keyword(s):  
Fundamental Problem ◽  
Logical Consequence ◽  
Logical Form ◽  
Logical Truth ◽  
Semantic Property ◽  
Truth Conditions ◽  
Property A ◽  
Logical Constants ◽  
Language L

A fundamental problem in the philosophy of logic is to characterize the concepts of ‘logical consequence’ and ‘logical truth’ in such a way as to explain what is semantically, metaphysically or epistemologically distinctive about them. One traditionally says that a sentence p is a logical consequence of a set S of sentences in a language L if and only if (1) the truth of the sentences of S in L guarantees the truth of p and (2) this guarantee is due to the ‘logical form’ of the sentences of S and the sentence p. A sentence is said to be logically true if its truth is guaranteed by its logical form (for example, ‘2 is even or 2 is not even’). There are three problems presented by this picture: to explicate the notion of logical form or structure; to explain how the logical forms of sentences give rise to the fact that the truth of certain sentences guarantees the truth of others; and to explain what such a guarantee consists in. The logical form of a sentence may be exhibited by replacing nonlogical expressions with a schematic letter. Two sentences have the same logical form when they can be mapped onto the same schema using this procedure (‘2 is even or 2 is not even’ and ‘3 is prime or 3 is not prime’ have the same logical form: ‘p or not-p’). If a sentence is logically true then each sentence sharing its logical form is true. Any characterization of logical consequence, then, presupposes a conception of logical form, which in turn assumes a prior demarcation of the logical constants. Such a demarcation yields an answer to the first problem above; the goal is to generate the demarcation in such a way as to enable a solution of the remaining two. Approaches to the characterization of logical constants and logical consequence are affected by developments in mathematical logic. One way of viewing logical constanthood is as a semantic property; a property that an expression possesses by virtue of the sort of contribution it makes to determining the truth conditions of sentences containing it. Another way is proof-theoretical: appealing to aspects of cognitive or operational role as the defining characteristics of logical expressions. Broadly, proof-theoretic accounts go naturally with the conception of logic as a theory of formal deductive inference; model-theoretic accounts complement a conception of logic as an instrument for the characterization of structure.

Second-order logic, philosophical issues in

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Expressive Power ◽  
Deductive System ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Structures ◽  
First Order ◽  
Correct Reasoning ◽  
Second Order Logic

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that range over sets, functions, properties or relations on the domain of discourse. A language is third-order if it has variables ranging over sets of sets, or functions on relations, and so on. A language is higher-order if it is at least second-order. Second-order languages enjoy a greater expressive power than first-order languages. For example, a set S of sentences is said to be categorical if any two models satisfying S are isomorphic, that is, have the same structure. There are second-order, categorical characterizations of important mathematical structures, including the natural numbers, the real numbers and Euclidean space. It is a consequence of the Löwenheim–Skolem theorems that there is no first-order categorical characterization of any infinite structure. There are also a number of central mathematical notions, such as finitude, countability, minimal closure and well-foundedness, which can be characterized with formulas of second-order languages, but cannot be characterized in first-order languages. Some philosophers argue that second-order logic is not logic. Properties and relations are too obscure for rigorous foundational study, while sets and functions are in the purview of mathematics, not logic; logic should not have an ontology of its own. Other writers disqualify second-order logic because its consequence relation is not effective – there is no recursively enumerable, sound and complete deductive system for second-order logic. The deeper issues underlying the dispute concern the goals and purposes of logical theory. If a logic is to be a calculus, an effective canon of inference, then second-order logic is beyond the pale. If, on the other hand, one aims to codify a standard to which correct reasoning must adhere, and to characterize the descriptive and communicative abilities of informal mathematical practice, then perhaps there is room for second-order logic.

Goldilocks an Engineer?

2016 ◽  
Vol 22 (6) ◽  
pp. 375-378
Author(s):  
Lukas J. Hefty
Keyword(s):  
Science Education ◽  
Fairy Tales ◽  
Mathematical Practice ◽  
Broad Sense ◽  
The Common ◽  
And Mathematics ◽  
Systematic Practice

How do you teach engineering to kindergartners? This is a fair question, given the stereotype of STEM workers as lab scientists and number crunchers; however, when approached from a wider perspective, even the youngest of children can be engineers. A framework for K–grade 12 science education defines engineering “in a very broad sense to mean any engagement in a systematic practice of design to achieve solutions to particular human problems” (NRC 2012, pp. 11–12). This aligns closely with the first of the Common Core's Standards for Mathematical Practice (SMP 1): Make sense of problems and persevere in solving them (CCSSI 2010, p. 6). Children as young as kindergarten are capable of identifying problems and designing solutions in science and mathematics, and it turns out that many fairy tales provide a rich problemsolution context.

Pragmatic truth and approximation to truth

1986 ◽  
Vol 51 (1) ◽  
pp. 201-221 ◽  
Author(s):  
Irene Mikenberg ◽  
Newton C. A. da Costa ◽  
Rolando Chuaqui
Keyword(s):  
Mathematical Formulation ◽  
Theory Of Knowledge ◽  
Major Influence ◽  
Mathematical Treatment ◽  
Starting Point ◽  
Pragmatic Truth ◽  
Classical Correspondence ◽  
Great Progress ◽  
And Mathematics

There are several conceptions of truth, such as the classical correspondence conception, the coherence conception and the pragmatic conception. The classical correspondence conception, or Aristotelian conception, received a mathematical treatment in the hands of Tarski (cf. Tarski [1935] and [1944]), which was the starting point of a great progress in logic and in mathematics. In effect, Tarski's semantic ideas, especially his semantic characterization of truth, have exerted a major influence on various disciplines, besides logic and mathematics; for instance, linguistics, the philosophy of science, and the theory of knowledge.The importance of the Tarskian investigations derives, among other things, from the fact that they constitute a mathematical, formal mark to serve as a reference for the philosophical (informal) conceptions of truth. Today the philosopher knows that the classical conception can be developed and that it is free from paradoxes and other difficulties, if certain precautions are taken.We believe that is not an exaggeration if we assert that Tarski's theory should be considered as one of the greatest accomplishments of logic and mathematics of our time, an accomplishment which is also of extraordinary relevance to philosophy, as we have already remarked.In this paper we show that the pragmatic conception of truth, at least in one of its possible interpretations, has also a mathematical formulation, similar in spirit to that given by Tarski to the classical correspondence conception.

Mathematics Based on Fuzzy Logic

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Mathematical Analysis ◽  
Classical Logic ◽  
Mathematical Reasoning ◽  
Narrow Sense ◽  
And Mathematics ◽  
Truth Degrees

Mathematical reasoning is governed by the laws of classical logic, based on the principle of bivalence. With the acceptance of intermediate truth degrees, the situation changed substantially. This chapter begins with a characterization of mathematics based on fuzzy logic, an identification of principal issues of its development, and an outline of this development. It then examines the role of fuzzy logic in the narrow sense for developing mathematics based on fuzzy logic and the main approaches developed toward its foundations. Next, some selected areas of mathematics based on fuzzy logic are presented, such as the theory of sets and relations, algebra, topology, quantities and mathematical analysis, probability, and geometry. The chapter concludes by examining various semantic questions regarding fuzzy logic and mathematics based on it.

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